Week1

Vectors in Two Dimensions

Vector Representation

Imagine a raft drifting down a river, where the speed and direction of the raft can be represented by an arrow. The arrow length represents the raft's speed, with longer arrows indicating greater speeds, while the arrow orientation indicates the direction in which the raft is headed.

Definition of Vectors

Vectors are quantities characterized by both length (magnitude) and direction. Examples include the motion of a raft in a river, electric and magnetic fields, and airflow over an airplane wing. Vectors are defined by both magnitude and direction, where two vectors are equal if they have the same magnitude and direction. Scalars are defined as quantities that possess only magnitude, without direction. A zero vector, denoted as 0, has zero length and no direction.

Scalar Multiplication

Given a scalar (c) and a vector (v), the scalar multiple (cv) represents a vector with its length equal to |c| times the length of v. If c is greater than 0, cv shares the same direction as v, whereas if c is less than 0, cv points in the opposite direction. Two vectors are parallel if they are scalar multiples of each other.

Vector Addition

Vector addition involves combining vectors, illustrated by considering a plane flying horizontally at a constant speed in a crosswind. The combined velocity is expressed as v = V + w, where v is the total velocity relative to the ground, V signifies the plane's airspeed, and w represents the wind's direction and speed. The methods for adding vectors include the Triangle Rule, where the tail of one vector is placed at the head of another, and the Parallelogram Rule, which forms a parallelogram to determine the resultant vector.

Vector Difference

The difference of two vectors, expressed as (u - v), can be written as (u + (-v)). This relationship can be visualized using the Triangle Rule, where the resultant vector extends from the tail of u to the head of v.

Vector Components

A position vector (v) with its tail at the origin and head at (v1, v2) is denoted as (v1, v2). The components of the vector are v1 (x-component) and v2 (y-component). For two position vectors to be equal, their corresponding components must also be equal.

Magnitude of Vectors

The magnitude of a vector can be determined using the formula |PQ| = √((x2 - x1)² + (y2 - y1)²) for points P(x1, y1) and Q(x2, y2).

Operations with Vectors

In two-dimensional space, three primary operations are performed: Addition, Subtraction, and Scalar Multiplication. Using component forms, these operations are expressed respectively as (u + v = (u1 + v1, u2 + v2)), (u - v = (u1 - v1, u2 - v2)), and (cu = (cu1, cu2)).

Unit Vectors

A unit vector is defined as any vector with a length of 1, with useful unit vectors being the coordinate unit vectors (i, j, k) defined as i = (1,0,0), j = (0,1,0), and k = (0,0,1).

Vectors in Three Dimensions

Many physical phenomena require three-dimensional representations, utilizing the xyz-coordinate system created by introducing the z-axis, which is perpendicular to the x and y axes. Points in three-dimensional space are described using ordered triples (x, y, z). In this context, the xy-plane contains all points with a z-coordinate of 0, represented by z = 0, while planes parallel to the xz-plane can be represented similarly using constant coordinates.

Spheres and Balls

A sphere centered at (a, b, c) with radius r is represented by the equation (x-a)² + (y-b)² + (z-c)² = r², while a ball is described by the inequality (x-a)² + (y-b)² + (z-c)² ≤ r².

Example Problems

Balancing forces involves calculating the forces exerted by chains suspending an engine at equilibrium, illustrated using a vector diagram leading to equations representing the forces. Additionally, working with given points to derive position vectors, computing lengths, and identifying unit vectors aids in understanding directionality in physics.

Applications of Vectors

An example of vector analysis in action is determining the speed and heading of a boat moving in a river, which involves a consideration of current dynamics and boat velocity.

Online Resources

Tools like GeoGebra and Desmos are useful for vector analysis, particularly for visualizing and performing calculations with vectors in multiple dimensions.